Max-Sugeno-Weber Composition Overview

• Max-Sugeno-Weber Composition is a family of aggregation operators that fuses Sugeno–Weber t-norms with max operations to address fuzzy optimization and multi-criteria decision challenges.
• It leverages algebraic properties such as t-norm dominance, pseudo-polynomial factorization, and order preservation to maintain consistency in complex fuzzy relational systems.
• Applications span fuzzy optimization, multi-criteria decision analysis, fuzzy relational equations, and game theory, providing robust tools for qualitative and non-linear data aggregation.

The Max-Sugeno-Weber Composition refers to a family of aggregation operators and compositional schemes that combine Sugeno–Weber t-norms with the max (supremum) operation, with applications spanning fuzzy optimization, multi-criteria decision, modeling under uncertainty, and systems of fuzzy relational equations. This composition leverages both the algebraic flexibility of the Sugeno–Weber family and the robust, qualitative aggregation provided by max-based structures.

1. Definition of Max-Sugeno-Weber Composition

The central ingredient of the Max-Sugeno-Weber (MSW) scheme is the Sugeno–Weber family of t-norms, $T^{\lambda}$, parameterized by $\lambda \in [0, \infty]$: $$T^{\lambda}(u, v) = \begin{cases} uv & \text{if } \lambda=0 \ \max\{0, (1-\lambda)uv + \lambda(u+v-1)\} & \text{if } 0 < \lambda < \infty \ \min\{u, v\} & \text{if } \lambda = \infty \end{cases}$$ This family interpolates between the product t-norm, nilpotent t-norms, and the minimum t-norm. In the "max–Sugeno–Weber" context, compositions typically involve aggregation functions such as the Sugeno integral, where the max (supremum over sets), min (infimum/meet), and the Sugeno–Weber t-norm play key roles.

A canonical MSW composition in fuzzy relational constraints or aggregation problems takes the form: $z(x) = \max_{j} T_\text{sw}(a_{ij}, x_j)$ for specified vectors or matrices $a$, $x$, and an appropriate value of $\lambda$. More general disjunctive normal forms and composite operators arise in multicriteria aggregation: $$f(x_1, \ldots, x_n) = \bigvee_{I \subseteq [n]} \left[ f(e_I) \wedge \bigwedge_{i \in I} \varphi_i(x_i) \right]$$ where the inner $f(e_I)$ are threshold/weight terms, $\varphi_i$ are monotonic local functions, and composition with a further maximum yields a max-dominant aggregation.

2. Algebraic Structure and Dominance Relations

Within the Sugeno–Weber family, dominance is defined as

$$T_1 \gg T_2 \iff \forall x, y, u, v \in [0,1], \quad T_1(T_2(x, y), T_2(u, v)) \geq T_2(T_1(x, u), T_1(y, v))$$

A complete characterization of when $T^{\lambda} \gg T^{\mu}$ is provided by a set of conditions involving $\lambda$, $\mu$, and critical values such as $17+12\sqrt{2}$, yielding a nontrivial partial order on the family. Notably, the dominance relation is transitive, which is not generally true for arbitrary t-norms (Kauers et al., 2010).

The use of the max operator and the transitive order structure allows composite operators like Max–Sugeno–Weber compositions to inherit desirable properties:

• Choice of $\lambda$ can be guided by how "strong" or "conservative" the aggregation should be.
• Transitivity enables layered compositions while maintaining aggregation strength ordering.
• Max-compositions preserve the dominance ordering, crucial for ensuring consistency in multi-stage systems.

3. Axiomatizations and Factorization of Max-Type Aggregations

Aggregation functions admitting a Max–Sugeno–Weber structure can be characterized by decomposability, pseudo-homogeneity, and order-preservation properties. In the general framework of Sugeno utility functions, pseudo-Sugeno integrals (pseudo-polynomial functions) are defined as those that can be uniquely factorized as

$$f(x_1, \ldots, x_n) = q(\varphi_1(x_1), \ldots, \varphi_n(x_n))$$

where $q$ is a Sugeno integral and each $\varphi_i$ is order-preserving.

A disjunctive normal form,

$$f(x_1, \ldots, x_n) = \bigvee_{I \subseteq [n]} [f(e_I) \wedge \bigwedge_{i \in I} \varphi_i(x_i)]$$

arises by Goodstein’s theorem. For functions with pronounced max-type behavior—the essence of the MSW composition—this normal form reflects a max-dominant structure parameterizable by the choice of $q$ and the thresholds $f(e_I)$. Such functions satisfy:

• Pseudo-median decomposability,
• Pseudo-min and pseudo-max homogeneity,
• Order preservation, ensuring they admit a factorization compatible with the underlying max–Sugeno–Weber design (Couceiro et al., 2011).

4. Computational and Structural Properties in Optimization

MSW compositions often appear as fuzzy relational inequality constraints: $$\max_j \left\{ T_\text{sw}(a_{ij}, x_j) \right\} \leq b_i$$ The feasible set for such problems is non-convex but can be described as a finite union of intervals/hypercubes $[x(e), X]$; each $x(e)$ is a minimal solution constructed via combinatorial selection according to which indices satisfy the underlying inequalities.

Key algorithmic steps:

• Feasibility: Determined by a sufficient condition: for all constraints, there exists some $j$ with $d_{ij} \geq b_i$ (Ghodousian et al., 16 Sep 2025).
• Solution structure: All minimal solutions are enumerated as $x(e)$ parameterized by suitable mappings $e$ (index selections); the feasible region is the union $[x(e), X]$ over all minimal $x(e)$.
• Optimization: The minimal value of the max-objective is achieved at one of these minimal solutions.
• Algorithm: Consists of (1) feasibility checking, (2) explicit construction of minimal solutions, (3) pairwise minimality comparison to remove dominated candidates, and (4) selection of those with minimal $z(x) = \max_i x_i$.

A numerical example with $n=10$ variables and $\lambda = 2$ demonstrates that the minimal solution set may be strictly less than the number of all candidates, and the global optimum is attained at these points (Ghodousian et al., 16 Sep 2025).

5. Learning and Identification via Fuzzy Relational Equations

When constructing or learning a Sugeno-integral (or its capacity function) to represent training data, the MSW composition naturally arises by expressing the empirical aggregation constraints as fuzzy relational equations using max–min products. Two key systems are formed for $N$ training samples, each represented as $(x^{(k)}, \alpha^{(k)})$: $$\begin{aligned} & \text{Max–min:} & \quad \max_A \min \{\min_{i\in A} x^{(k)}_i, \mu(A)\} = \alpha^{(k)} \ & \text{Min–max:} & \quad \min_A \max \{\max_{i\notin A} x^{(k)}_i, \mu(A)\} = \alpha^{(k)} \end{aligned}$$ Sanchez’s results yield extremal (greatest/lowest) solutions, with the maximal solution characterized by explicit application of the Gödel implication to transfer from data to capacity parameters.

Reductions can be made (e.g., to $q$-maxitive/minitive capacities) by focusing only on sets $A$ of limited cardinality. For noisy or inconsistent data, Chebyshev distance-based approximations are constructed to minimize the perturbation needed to restore consistency, and the resulting best-approximate capacities are identified accordingly (Baaj, 14 Aug 2024).

6. Robustness, Compatibility, and Generalizations

Max–Sugeno–Weber compositions inherit and rely on several structural properties:

• Max–min structure: The interleaving of max and min operations (in the integral’s or composition’s formula) provides robustness to imprecision and interval-valued data, as formalized by robust Sugeno integrals (Greco et al., 2012).
• Compatibility with congruences: Sugeno integrals (and hence MSW compositions) uniquely preserve congruences on their domain; thus, any aggregation compatible with all lattice congruences must be a Sugeno integral (Halaš et al., 2018).
• Comonotonicity properties: New axiomatizations for the Sugeno integral via generalized comonotonicity on lattices facilitate computational efficiency and generalize the preservation of maxima/minima across composed operators (Halaš et al., 2018).
• Scale invariance: MSW compositions naturally preserve aggregation properties under scale transformations and coarsenings, which is useful in ordinal and qualitative decision support frameworks (Halaš et al., 2018).
• Abstract convexity: In game-theoretic applications, where the Sugeno integral is used for payoff modeling in the presence of qualitative or non-additive uncertainty, abstract convexity structures enable fixed-point existence and generalize the existence of Nash equilibria (Radul, 2015).

7. Applications and Practical Significance

The Max-Sugeno-Weber composition appears in several domains:

• Fuzzy optimization: In nonlinear programs with fuzzy relational constraints, especially where both the aggregation (via max) and the underlying logic (via Sugeno–Weber t-norms) reflect non-additive uncertainty, exact optima can be found by exploiting the structure of the feasible set as a finite union of intervals and relations among minimal solutions (Ghodousian et al., 16 Sep 2025).
• Learning from data: Systems of fuzzy relational equations leveraging MSW compositions enable capacity learning for Sugeno integrals, including greatest and lowest (extreme) representing capacities, reduction to $q$-maxitive capacities, and Chebyshev-approximate solutions for inconsistent data (Baaj, 14 Aug 2024).
• Robust decision analysis: When input evaluations are interval-valued or imprecise, robust Sugeno integrals generalize the classical MSW composition to compute a one-shot, qualitatively meaningful aggregate (Greco et al., 2012).
• Game theory: The maximum-based structure underlies the modeling of payoffs with Sugeno integrals in qualitative games, ensuring the preservation of convexity-like properties and Nash equilibrium existence even in categorical, non-linear settings (Radul, 2015).

The MSW composition thus provides a unifying algebraic and algorithmic framework that encompasses a range of qualitative aggregation, decision, and learning problems. Its theoretical foundations—rooted in properties of t-norm dominance, lattice compatibility, pseudo-polynomial factorization, and robustness to imprecision—enable principled design and analysis of aggregation operators across multicriteria, optimization, and uncertainty-aware settings.